Excerpt from the Proceedings of the COMSOL Conference 2009 Boston

MultiPhysics Analysis of Trapped in Multi-Layer YBCO Plates

Philippe J. Masson1 1Advanced Lab *1720 Main Street, Bldg. #4, Palm Bay, Fl-32905, [email protected]

Abstract: superconductors have the unique configuration specifically developed for this capability of trapping magnetic flux. This feature specific application. The paper presents a FEA has the potential to enable and improve several electromagnetic-thermal analysis of flux trapping applications including high power density in YBCO multi-layers plates using COMSOL rotating machines. Current material used as MultiPhysics. trapped flux (TFM) is single domain YBCO that present numerous limitations in 2. Flux trapping in superconductors terms of performance, stability and size. One way to overcome the limitations is to use thin Because of their unique properties, layers of YBCO deposited on disks and stack superconductors can trap magnetic flux and act them. Multi-layer trapped flux magnets were as “permanent” magnets. The phenomenon can successfully modeled and analyzed using be explained by the critical state model in which COMSOL MultiPhysics through coupled current density can only be ±Jc or 0. Since electromagnetic and thermal transient of superconductors is hysteretic in simulations. Simulations allow for a better nature, the magnetic state of the material understanding of how the current redistributes depends of the and external field during thermal disturbances and validate the history. potential of the technology. Keywords: trapped flux magnets, 2.2 Flux trapping methods superconductors, YBCO, non-linear problem Several methods to trap magnetic flux in a 1. Introduction superconductor can be used. Their effectiveness and practicality differ from one method to Trapped Flux Magnets (TFM) are very another. attractive for application to power devices. The • Field cooling major limitation of flux trapping capabilities of bulk YBCO stems from mechanical problems. Field cooling consists of a cool down of the Indeed, Lorenz forces can become very superconductor under applied field. This method important when stored magnetic is very effective but requires a large magnet increases. As most of the stress is applied locally generating at least the value of flux density on pinning centers in the material, a failure desired in the superconductor. usually results in destruction of the material structure. In order to improve flux trapping • Zero field cooling capability in TFMs, mechanical reinforcement can be done using epoxy impregnated fiber glass If zero-field cooling is performed, the cloth and a record of 17 T trapped at 29 K was superconducting material will first react by achieved. However, due to the ceramic nature of shielding flux variation and will expel magnetic YBCO, it remains structurally weak and can very flux from its volume. The method then requires likely fail if electro-thermal instabilities occur. full saturation in current and necessitates large We have studied the feasibility of TFMs based magnets providing at twice the value of flux on multi-layer configurations that could be density desired in the superconductor. achieved by stacking disks made of coated conductors. The multi-layer configuration would • Pulsed magnetization bring a much better stress distribution and should lead to more stable flux trapping. Even though Pulse magnetization is identical in principle to packing factor remains an issue for stacked zero field cooling, however, because the flux coated conductors, it is expected that YBCO can variation is very fast, the superconductor can be deposited advantageously in multiple layer enter a flux flow regime and generate losses decreasing the magnitude of the critical current the super conductor without adding weight or density. Additionally, the method r equire s volume to the machine. An example of such energy storage for pul se generation . machine is shown in figure 1 [2]. The machine takes advantage of the flux trapping capability of • Flux pumping superconducting plates to create, shape and concentrate magnetic flux leading to a very high Recently flux pumping techniques have been power density. developed requiring a c omplex to set up , controllable temperature and magnetic gate material . This method represents a very promising alternative as no large field source is needed.

Field cooling is the most eff ective way to trap magnetic flux in a superconducting material. After a field cooling, a superconducting plate behaves like a permanent source of . Its major differences with permanent magnets are: Figure 1. Illustration of Bean’s model • a much larger magnetization • operation at constant flux instead of 4. Properties of YBCO constant magnetization (trapped flux magnets cannot be “defluxed”) The most widely used material in TFMs is YBCO. YBCO is a ceramic that become s Table 1: Field cooling of a superconducting plate Step 1 Step 2 Step 3 superconducting below 92 K and exhibits very high current density in single domain blo cks. A Bext = B max Bext = B max Bext = 0 T > T c T < T c T < T c flux density of 17 T has been trapped in a small YBCO plate at 29 K [3]. While bulk YBCO is a very attractive option, it presents severe limitations: • Brittle material that can develop cracks • Stress applied by the Lorentz forces on

pinning centers ca n lead to mechanical 3. Applications of TFMs failure

• Size limited to a few centimeters with Trapped flux magnets enable the development of homogeneous properties several superconducting devices requiring • Cannot be bent constant magnetic flux or large values of magnetic flux density. • Magnetic bearings Some applications, such as flywheels or high speed electrical machines, require frictionless bearings. TFMs can be advantageously used as they can provide very large forces and do not require active control. Indeed, since the magnetic flux intrinsically re mains constant, any variation Figure 2. Single grain YBCO c ylinder (www.can.cz) is automatically compensated by induced persistent currents. Superconductors exhibit a highly non -linear • Rotating machines electrical conductivity as shown in figure 3 and High specific power rotating machine require equation (3). large excitation field that can be produced by TFMs [1] . The major challenge is to magnetize n(B,T ) •  J  Improved   • Larger systems can be built E = Ec   (3)  J c ()B,T 

5.00E-04 4.50E-04 4.00E-04 3.50E-04 3.00E-04 2.50E-04

E (V/m) 2.00E-04 1.50E-04 1.00E-04 5.00E-05 Figure 3. YBCO conductor from American 0.00E+00 50 70 90 110 130 150 Superconductor Inc. [4] I (A) Commercially available wide conductors can be used, Figure 3. E(I) characteristic of superconductors however, the stabilization layers necessary for stable conductor operation are not requires in TFM and a The critical current density of YBCO is higher filling factor can be achieved with a dedicated depending on the temperature and on the applied configuration. magnetic flux density. It can be parameterized as follows: 6. Problem definition B −  T   1  90 1−  6.1 Geometry    TB  J c (B,T ) = J c )0,0( 1− e (4)  Tc  The system is modeled in 2D with axial

The parameters J c(0 T, 0 K), T c and T b can be symmetry, which is a valid approximation for the obtained experimentally. Current flows in the a-b system of interest since YBCO can be deposited plane of the material and current density in the c- uniformly. The geometry implemented in axis of the material is negligible. COMSOL is shown in figure 4. Axis Field coil

5. Multilayer configuration TFM YBCO YBCO can be deposited in thin films and turned Epoxy resin into tapes. YBCO is the basis for second generation conductors or coated conductors and can be deposited on large areas. Up to 10 centimeter wide ribbons are currently manufactured and cut into 4 mm wide tapes to make conductors. Disks coated with YBCO film can be manufactured and stacked to form Heater cylinders. The electrical properties would then be Figure 4. Geometry implemented in Comsol strongly anisotropic as current is allowed to flow only in parallel planes (a-b planes) and no The system is modeled with homogeneous YBCO current is flowing between layers. YBCO is layers 0.15 mm thick and separated by 0.1 mm layers deposited on substrate and the filling factor of a of resin insulation. stack is a lot less than 50 %. However, current density in YBCO films or thin layers is orders of 6.2 Material properties magnitude larger than in large block and therefore compensate for the low filling factor. • YBCO Such a configuration brings numerous advantages: YBCO exhibits non-linear electrical and thermal • Material can be bent properties. In COMSOL, conductors are • Intrinsically mechanically reinforced modeled with their electrical conductivity, based on equation (3), the conductivity of YBCO can 6.3 Mesh, Sources and boundary conditions be expressed as follows. The mesh, represented in figure 7, is composed 3   1 2   −1     2   n of about 80,000 elements leading to a system J c0  T   1  E σ[S / m] = 1−     with over 300,000 degrees of freedom. The mesh E   T    B  E  c  c  1+  c is kept coarse in the surrounding air and in the B  0  (5) field coil.

Where T c is the critical temperature, J c0 is the value of the critical current density at 0 K and 0 T, E c is the electrical field magnitude defining Jc, B 0 and n are determined experimentally. The electrical field E is a parameter of the electrical conductivity which would pose a problem in most FEA package but COMSOL allows for this dependence to be modeled.

Specific and thermal conductivity of YBCO Figure 7. Mesh in the multi-layer component are plotted as a function of temperature in figure 5. The thermal simulation only considers the YBCO, resin and heater. A heat exchange Thermal properties of YBCO condition is set on the resin simulating cooling. 350 4 Cp YBCO K YBCO The heat pulse applied to the heater is shown in 300 3.5

3 figure 8. 250 k (W/(m.K) 2.5 200 2 150 1.5 Cp (J/(kg.K) Cp 100 1

50 0.5

0 0 50 70 90 110 130 150 Temperature (K)

Figure 5. Thermal properties of YBCO

• Resin

G10 resin is used to hold the YBCO layers together. Its thermal properties are shown in figure 6. Figure 8. Heat pulse.

Thermal properties of resin The field coil generates a flux density on the 450 0.8 Cp resine TFM which is ramping down linearly from 5 T 400 0.7 k resine 350 to 0 T. 0.6 300

0.5 k (W/(m.K) 250 0.4 7. Simulation sequence 200 0.3

Cp (J/(kg.K) Cp 150 0.2 100 Since field cooling requires full field penetration 50 0.1 before the YBCO becomes superconducting, the 0 0 50 70 90 110 130 150 initial condition corresponding to this state needs Temperature (K) to be computed first. When a superconductor is cooled down under field, part of the magnetic Figure 6. Thermal properties of G10 resin flux is expelled from its volume; if the applied field is below H c1 , which is of a few mT, then the material is diamagnetic (), if the Table 2: current penetration in the TFM applied field is greater than Hc1, then the 1 material is in the mixed state and normal zones are developed allowing quanta of flux to go through the material. In the case presented, the material is in the mixed state and the resulting partial which is typically of a few percents is neglected. The simulation sequence is the following: 1. Initial state (steady state): maximum external field applied, T=85 K 2 2. Transient analysis part 1: external field ramped down, no external heat source 3. Transient analysis part 2: no applied field, apply pulse of heat

8. Simulation Results

Once the flux is trapped, the TFM is excited with 3 a heat pulse. This thermal disturbance can lead to a decrease of the trapped flux or a complete demagnetization of the material. A critical value for the heat pulse leading to non-stable behavior can be determined. This relates to the minimum quench energy (MQE) for quench studies which is the minimum amount of energy leading to instability of the material. The two cases are described in this section.

8.1 Current penetration 8.2 Stable behavior

As predicted by the critical state model, the A heat pulse of 18 J is applied to the TFM once current induced by a decreasing magnetic flux the field coil is completely ramped down. As penetrates from the side in the material. Since all shown in following table, the current in the the layers are magnetically coupled, the stack layers closer to the heater heat up, thus becoming behaves as a single block of YBCO. The dissipative. The heat exchange is high enough to magnitude of the current density strongly maintain the top layers in the superconducting depends on the applied flux density leading to a state; the trapped flux is nearly unchanged. lower current value in the central part of the Indeed, when the current in the bottom layer is TFM. dissipated, the associated flux variation is In table 2: compensated by a redistribution of the current in 1- The applied field ramp-down starts, the top layers; the current penetrates deeper in current penetrates in the superconductor the material. In table 3: 2- As the field keeps decreasing, current 1- Current starts penetrating in the material penetration increases, current density is 2- Magnetic flux is trapped already lowered in the center of the 3- Heat pulse is generated decreasing TFM current magnitude in the lower layers 3- All the field is generated by the TFM 4- After current redistributes to keep flux variation to a minimum The current can penetrate until the material is fully saturated for larger applied field values.

Table 3: stable behavior – Color represents current field coil. The end of the simulation shows the density and lines the flux lines. field trapped in the superconductor.

1

2

Figure 9. flux distribution on top of the TFM

During flux trapping, magnetic flux is changing in the area where current is flowing. Any change of flux creates an electrical field as predicted by Faraday’s law of induction. The simultaneous presence of electrical field and current density creates losses. Such losses can be calculated by integrating J×E over the volume of 3 superconductor. The instantaneous power, shown in figure 10, peaks at 0.5 mW and leads to a 0.2 mJ of dissipated energy in the material, which in this case is negligible. However, for faster flux variation, these losses can become significant.

4

Figure 10. Heat dissipated in the superconductor

8.3 Unstable behavior

Figure 9 shows the flux distribution on the top of In some cases, an avalanche phenomenon can the TFM during the first 10 s. The initial occur destroying the magnetization of the distribution is the uniform field generated by the material. This can happen for large energy deposition such as heat pulse and/or weak cooling. In the presented simulation, the heat pulse was increased by 20 % leading to instability. The following table shows the simulation results.

Table 4: unstable behavior – Color represents current density and lines the flux lines. 1

Figure 11. Flux distribution the center top of the TFM

2

3 Figure 12. Heat dissipated in the superconductor

9. Conclusion

Multi-layer trapped flux magnets present a very interesting alternative to bulk blocks of ceramic YBCO. A model was developed to investigate their stability against thermal disturbances. Modeling superconducting material is not trivial

because of the strong non-linearity of the In table 4: electrical conductivity. COMSOL allows for a 1- No external field. The flux is trapped better understanding of the behavior of trapped and the heat pulse starts flux magnet through successful coupled 2- Current redistributes and penetrates electromagnetic and thermal simulations. deeper in the material

3- An avalanche phenomenon dissipates 10. References all the stored energy 4- 1. P. Masson et Al., Design of HTS Axial Flux IEEE Trans. The magnitude of the flux density at the center Motor for Aircraft Propulsion”, Appl. Supercon. 17 top of the TFM is shown in figure 11. The entire , , 2, pp 1533-1536, (2007) trapped field disappears at the beginning of the 2. P. Masson et Al., High Power Density pulse, evidence of the instability. Figure 12 Superconducting Motor For All-Electric Aircraft Trans. Appl. Supercon 15 shows the energy dissipated in the Propulsion, ., , 2, 2226- superconductor. A peak of 0.25 W appears at the 2229, (2005) beginning of the pulse releasing the stored 3. Tomita et Al., High-temperature energy. superconductor bulk magnets that can trap magnetic fields of over 17 at 29 K, Nature , 421 , 517-520, (2003) 4. www.amsuper.com